Factor x^3 - 4x^2 + x + 6 with all steps
let f(x) = x^3 - 4x^2 + x + 6
f(1) = 1-4+1+6 ≠0
f(-1) = -1 - 4 -1 + 6 = 0
so (x+1) is a factor
Using synthetic division or long algebraic division
x^3 - 4x^2 + x + 6
= (x+1)(x^2 - 5x + 6)
= (x+1)(x-6)(x+1)
so x = 6 or x = -1 (a double root at x=-1)
thank you!
To factor the given polynomial, we will use a method called "grouping." Here are the steps to factor the polynomial x^3 - 4x^2 + x + 6:
Step 1: Look for a common factor, if any. In this case, there is no common factor other than 1.
Step 2: Group the terms in pairs. In this case, we can group the first two terms (x^3 - 4x^2) and the last two terms (x + 6).
So, our polynomial can be rewritten as (x^3 - 4x^2) + (x + 6).
Step 3: Factor out the greatest common factor from each group.
From the first group, we can factor out x^2: x^2(x - 4).
From the second group, we can factor out 1: 1(x + 6).
Rewriting the polynomial, we have: x^2(x - 4) + 1(x + 6).
Step 4: Check if there are any common factors left. In this case, there are no further common factors.
Step 5: Combine the factored terms. The factored form of the given polynomial is: x^2(x - 4) + 1(x + 6).
And that's how you factor x^3 - 4x^2 + x + 6 using the grouping method!